The link describes the 3 different places you can find the serial number on. The charging case, on an iPhone, or on the original box. It may also be shown in the purchase receipt for them if you still have it.
Does your Galaxy Note not offer it in the Bluetooth settings under Previously connected devices? It should show the paired Airpods even if they are not currently connected, but may or may not show the serial number in the info page for them.
If you can't use any of the 3 options in the support document and your Android phone does not show it anywhere, then there's unfortunately no other way to get the serial number.